A062196 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(\left(6\,x + 24\,n + 2\,n x + 6\,n x^{2} + 44\,n^{2} - 6\,n^{2} x + 11\,n^{2} x^{2} + 24\,n^{3} - 2\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + n^{4} x^{2}\right)\,P_{n}(x) = \left(12\,x + 18\,x^{2} + 24\,n + 60\,n x + 24\,n x^{2} + 18\,n x^{3} + 44\,n^{2} + 92\,n^{2} x + 13\,n^{2} x^{2} + 29\,n^{2} x^{3} + 24\,n^{3} + 58\,n^{3} x + 4\,n^{3} x^{2} + 16\,n^{3} x^{3} + 4\,n^{4} + 12\,n^{4} x + n^{4} x^{2} + 3\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(12\,x + 6\,x^{2} - 18\,x^{3} + 44\,n x - 30\,n x^{2} - 6\,n x^{3} - 8\,n x^{4} + 36\,n^{2} x - 34\,n^{2} x^{2} + 6\,n^{2} x^{3} - 8\,n^{2} x^{4} + 8\,n^{3} x - 8\,n^{3} x^{2} + 2\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	3								5
1	2	11	6							19
2	2	23	36	10						71
3	2	39	120	90	15					266
4	2	59	300	430	190	21				1002
5	2	83	630	1470	1225	357	28			3795
6	2	111	1176	4060	5460	2982	616	36		14443
7	2	143	2016	9660	19110	16758	6468	996	45	55198

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-137a24c9ba6939cb · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-8233919c3a8ecd7d

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 2,   3;
  1: 2,  11,    6;
  2: 2,  23,   36,    10;
  3: 2,  39,  120,    90,     15;
  4: 2,  59,  300,   430,    190,     21;
  5: 2,  83,  630,  1470,   1225,    357,     28;
  6: 2, 111, 1176,  4060,   5460,   2982,    616,     36;
  7: 2, 143, 2016,  9660,  19110,  16758,   6468,    996,    45;
  8: 2, 179, 3240, 20580,  56280,  72324,  44688,  12840,  1530,   55;
  9: 2, 219, 4950, 40260, 145530, 257796, 232848, 106920, 23760, 2255, 66;

%F Formula

The row polynomials satisfy (4*n^4+24*n^3+44*n^2+24*n-(2*n^3+6*n^2-2*n-6)*x+(n^4+6*n^3+11*n^2+6*n)*x^2)*P(n,x) = (4*n^4+24*n^3+44*n^2+24*n+(12*n^4+58*n^3+92*n^2+60*n+12)*x+(n^4+4*n^3+13*n^2+24*n+18)*x^2+(3*n^4+16*n^3+29*n^2+18*n)*x^3)*P(n-1,x) + ((8*n^3+36*n^2+44*n+12)*x-(8*n^3+34*n^2+30*n-6)*x^2+(2*n^3+6*n^2-6*n-18)*x^3-(2*n^3+8*n^2+8*n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^4 + 24*n^3 + 44*n^2 + 24*n + (12*n^4 + 58*n^3 + 92*n^2 + 60*n + 12)*x + (n^4 + 4*n^3 + 13*n^2 + 24*n + 18)*x^2 + (3*n^4 + 16*n^3 + 29*n^2 + 18*n)*x^3)*P[n - 1, x] + ((8*n^3 + 36*n^2 + 44*n + 12)*x - (8*n^3 + 34*n^2 + 30*n - 6)*x^2 + (2*n^3 + 6*n^2 - 6*n - 18)*x^3 - (2*n^3 + 8*n^2 + 8*n)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 24*n^3 + 44*n^2 + 24*n - (2*n^3 + 6*n^2 - 2*n - 6)*x + (n^4 + 6*n^3 + 11*n^2 + 6*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^4 + 24*n^3 + 44*n^2 + 24*n + (12*n^4 + 58*n^3 + 92*n^2 + 60*n + 12)*x + (n^4 + 4*n^3 + 13*n^2 + 24*n + 18)*x^2 + (3*n^4 + 16*n^3 + 29*n^2 + 18*n)*x^3)*v[i - 1] + ((8*n^3 + 36*n^2 + 44*n + 12)*x - (8*n^3 + 34*n^2 + 30*n - 6)*x^2 + (2*n^3 + 6*n^2 - 6*n - 18)*x^3 - (2*n^3 + 8*n^2 + 8*n)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 24*n^3 + 44*n^2 + 24*n - (2*n^3 + 6*n^2 - 2*n - 6)*x + (n^4 + 6*n^3 + 11*n^2 + 6*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-039745f356dca7e2

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

obreschkoff_combination e-039745f356dca7e2

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-32e7e43d5a9debda root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.